Give a recursive definition of the set of positive integer powers of 3. That is the set {3,9,27,81,...}

Explanation:

From the given information:

Consider the first positive integer power of 3 as 3¹ = 3

SO; 3 ∈ S

Thus, for every positive integer power of 3, it is set to be the previous integer of 3 multiplied by 3.

i.e. 3s ∈ S   if and only if s ∈ S

Now for the first iteration, the recursive definition on s = 3

In the second iteration;

3s = 3 × 3 = 9 ∈ S

Thus, we use the recursive definition on s = 9 in the second iteration.

In the third iteration;

3s = 3 × 9 = 27 ∈ S

Thus, we use the recursive definition on s = 27 in the third iteration.

Then; 3s = 3 ×  27 = 81  ∈ S and so on.

Therefore, the set is {3,9,27,81,...}

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